Introduction For a finite group, one can evaluate a character on the sum of all m-th powers of the group elements. The resulting number, divided by the order of the group, is called the m-th Frobenius-Schur indicator of the character. The first use of these indicators was made by F. G. Frobenius and I. Schur (cf. [15]) to give a criterion when a representation of a finite group can be realized by matrices with real entries—for this question, it is the second indicator that is relevant. This is also meaningful for other fields than the complex numbers: Here the indicator tells whether or not a given module is self-dual. Higher indicators, i.e., indicators with m 2, arise when one considers the root number function in a finite group. This function assigns to a group element the number of its m-th roots, i.e., the number of group elements whose m-th power is equal to the given element. It is clear that this number depends only on the conjugacy class, and therefore defines a class function that can be expanded in terms of the irreducible characters. Using the orthogonality relations for characters, it is not hard to see that the coefficient of an irreducible character in this expansion is its m-th Frobenius-Schur indicator (cf. [22], Lem. (4.4), p. 49). For Hopf algebras, Frobenius-Schur indicators were first considered by V. Linchen- ko and S. Montgomery on the one hand (cf. [31]) and by J. Fuchs, A. Ch. Ganchev, K. Szlachanyi, and P. Vecsernyes on the other hand (cf. [16]). Here, the sum of the m-th powers of the group elements is replaced by the m-th Sweedler power of the integral. The authors then use the indicators, or at least the second indicator, to prove an analogue of the criterion of Frobenius and Schur whether or not a representation is self-dual: The Frobenius-Schur theorem asserts that this depends on whether the second indicator is 0, 1, or — 1. The topic of the present writing are the higher Frobenius-Schur indicators for semi- simple Hopf algebras and their relation to other invariants of irreducible characters. These other invariants are the order, the multiplicity, the exponent, and the index. Let us briefly describe the nature of these invariants. The notion of the order of an irreducible character is a generalization of the notion of the order of an element in a finite group: It is the smallest integer such that the corresponding tensor power contains a nonzero invariant subspace. The dimension of this invariant subspace is called the multiplicity of the irreducible character. An irreducible character has order 1 if and only if it is trivial, and has order 2 if and only if it is self-dual. In these cases, the multiplicity of the character is 1. l

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